Spearman's rank correlation for a bivariate sample $\{ (x_1, y_1), (x_2, y_2) , \ldots , (x_n, y_n) \}$ is generally defined as the correlation between the ranks of the observations, but what is the population analog of this? I think I recall seeing that in the continuous case at least it can be thought of as $\text{Corr}[F(X), G(Y)]$ where $F$ and $G$ are the distribution functions of $X$ and $Y$. This makes some sense since in calculating the correlation between the ranks in the sample case you could always divide through by $n$ and be looking at the values of the empirical distribution functions. In any event, is there another interpretation? Can it somehow be defined at the distribution level as something other than a correlation?
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There's an interpretation given in some work on copulas.
e.g. see p 15 of Embrechts et al (2001) [1], which has for the Spearman correlation of $(X,Y)^T$:
$\rho_S(X,Y)=3(\mathbb{P}\{(X-\tilde{X})(Y-Y')>0\}-\mathbb{P}\{(X-\tilde{X})(Y-Y')<0\})$
where $(X, Y)^T$, $(\tilde{X},\tilde{Y})^T$ and $(X',Y')^T$ are independent copies. (It then goes on to show your interpretation holds for that definition.)
[1] Paul Embrechts, Filip Lindskog and Alexander McNeil (2001),
"Modelling Dependence with Copulas and Applications to Risk Management"
http://www.risklab.ch/ftp/papers/DependenceWithCopulas.pdf
(alternative link)
